The
Reuleaux tetrahedron can be modified into a 'solid of constant width'
by replacing all six edges with sections of an 'envelope of spheres' -
a sphere-sweep in ray tracing terminology. The resulting solid is
spheroform and has tetrahedral symmetry.

The
sphere envelopes extend between vertices on each edge of the unit
tetradedron (edge length 1). The radius of each sphere in the envelope
is given by the function:

Each
sphere is tangent to the straight edge of an internal regular
tetrhadron, and tangent to the surface of the Reuleaux tetrahedron.